Optical systems and methods employing adjacent rotating cylindrical lenses

ABSTRACT

An optical system includes a pair of adjacent cylindrical lenses and a target positioned in a round-spot plane following the pair of cylindrical lenses. At least one of the cylindrical lenses is rotatable about the optical axis, relative to the other cylindrical lens. A collimated light beam is incident on the pair of cylindrical lenses. Rotating the cylindrical lenses relative to each other allows changing the size of a round spot (or symmetrically scaling an image) at a given target location. Additional optical elements, for example a spherical lens, may be placed after the pair of cylindrical lenses. In various embodiments, the pair of cylindrical lenses may include two positive lenses, or a positive and a negative lens.

RELATED APPLICATION DATA

This application is a continuation-in-part of U.S. patent applicationSer. No. 10/143,215, filed on May 9, 2002, now U.S. Pat. No. 6,717,745,which claims the benefit of the filing date of U.S. Provisional PatentApplication No. 60/290,157, filed May 9, 2001, both of which are hereinincorporated by reference.

FIELD OF THE INVENTION

The invention relates to optical systems and methods for magnifyingimages or controlling the sizes of laser spots or optical images, and inparticular to optical systems and methods employing rotating cylindricallenses and/or mirrors.

BACKGROUND OF THE INVENTION

Microscopes, image projectors, industrial laser optical systems, andvarious other optical systems have been used to control the sizes ofimages at a target location. Many such optical systems employ lensesand/or mirrors which are translated relative to each other along theoptical axis of the system in order to control the output image size. Insome systems, changing the size of the image alters the location of theplane at which the image is in focus. Moreover, some conventionalsystems can be relatively bulky, and require complex mechanicalcomponents.

In the article “Optical System for Image Rotation and Magnification,” J.Opt. Soc. Am. 70(2), February 1980, Braunecker et al. describe anoptical system including a pair of rotatable, adjacent cylindricallenses having focal lengths of equal magnitude and opposite signs.Braunecker et al. further show an additional, non-adjacent sphericallens situated away from the pair of cylindrical lenses.

SUMMARY OF THE INVENTION

The present invention provides an optical system comprising a firstcylindrical optical element for receiving an input light beam, and asecond cylindrical optical element positioned at a round spot locationoptically subsequent to the first cylindrical optical element. At leastone of the cylindrical optical elements is rotatable about the opticalaxis so as to adjust an angle between the principal axes of thecylindrical optical elements to symmetrically scale a light beam spot ata target location.

Further provided is an optical method comprising: generating asymmetrically-scalable spot on a target positioned at a working locationby passing a light beam sequentially through a first cylindrical opticalelement and a second cylindrical optical element, the second cylindricaloptical element being separated from the first cylindrical opticalelement by a distance chosen such that an input beam forms a circularspot at the second cylindrical optical element after passing through thefirst cylindrical optical element; and symmetrically scaling the spot atthe working location by adjusting an angle between a principal axis ofthe first cylindrical optical element and a principal axis of the secondcylindrical optical element by rotating at least one of the firstcylindrical optical element and the second cylindrical optical element.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing aspects and advantages of the present invention willbecome better understood upon reading the following detailed descriptionand upon reference to the drawings where:

FIG. 1 is a schematic diagram of an optical system comprising twocylindrical lenses, according to a presently preferred embodiment of thepresent invention.

FIGS. 2-A and 2-B are schematic diagrams illustrating the passage of alight beam through the system of FIG. 1, for two extreme angular lenspositions, respectively.

FIG. 3 illustrates another system configuration according to the presentinvention.

FIG. 4 shows another system configuration according to the presentinvention.

FIG. 5 shows yet another system configuration according to the presentinvention.

FIGS. 6-A and 6-B are schematic diagrams illustrating the passage of alight beam through yet another system configuration according to thepresent invention, for two extreme angular lens positions, respectively.

FIG. 7 is a schematic diagram of a system configuration including acylindrical lens and a cylindrical mirror, according to anotherembodiment of the present invention.

FIG. 8 is a schematic diagram of a system configuration including twocylindrical mirrors, according to another embodiment of the presentinvention.

FIG. 9 shows another configuration including two adjacent cylindricallenses, according to another embodiment of the present invention.

FIG. 10 shows yet another configuration including two adjacentcylindrical lenses, according to another embodiment of the presentinvention.

FIG. 11 illustrates in a longitudinal sectional view an implementationof mechanical components suitable for rotating one cylindrical lens withrespect to another, according to an embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

In the following description, it is understood that, unless explicitlyspecified otherwise, all recited distances are optical distances, whichmay or may not be equal to physical (free space) distances betweenoptical elements. The terms “round spot” and “circular spot” are notlimited to spots of uniform irradiance/intensity, and are understood toencompass modulated, symmetrically-scaleable images encodingpatterns/pictures. Such images may be projected on a target screen forexample. A “round spot” position of an optical element of a system isunderstood to be a position at which an input round spot is transformedto an output round spot by the part of the optical system opticallypreceding the optical element. Adjusting angles between the cylindricalaxes of optical elements refers to adjusting optical angles, i.e. anglesbetween the cylindrical axis of the elements and the direction of thetransverse (vertical or horizontal) axis at the cylindrical elementlocation. The statement that a second optical element is rotatable withrespect to a first optical element is understood to encompass rotatingeither or both of the optical elements.

The following description illustrates embodiments of the invention byway of example and not necessarily by way of limitation.

The general terminology and concepts employed in the ensuing descriptionare discussed below.

Terminology and Concepts

An optical system comprising spherical and cylindrical lenses and/ormirrors, and sections of free spaces, can be described mathematically bya 4×4 real matrix called the ABCD matrix. For simplicity, such a systemcan be considered to be made of centered, aligned, lossless, passive,and aberration-free elements. The ABCD matrix has specific mathematicalproperties (called symplecticity) which can limit the maximum number ofthe independent matrix elements to ten. In a first order approximation,most optics belong to this category. These optical systems are calledABCD-type systems. The beams propagate aligned to the ABCD-type opticalsystems.

A beam is a distribution of light in free space that does not depart toofar away from a reference axis. Such a distribution can be termed aparaxial distribution. The reference axis coincides with the beam axis.The beam axis is rigorously determined by using the first-order momentsof the beam irradiance (sometimes termed “intensity”). For consistency,in the following discussion the z-axis is the beam axis, and the x- andy-axes represent the horizontal and the vertical transverse laboratoryaxes, respectively.

The main physical quantities characterizing the simplest beam can bedefined using the second-order moments of the beam irradiance. They are,according to the International Organization for Standardization (ISO):the beam waist radius w₀ (half-diameter), the beam waist size (fulldiameter) D₀, the waist location z₀, the beam divergence (full angle) θ,the Rayleigh range Z_(R), (Z_(R)=D₀/θ), and the beam propagationparameter M². The beam propagation parameter is also named thetimes-diffraction-limit factor. These quantities are defined for astigmatic (ST) beam, which is rotationally-symmetric about the z-axis,and forms a round spot everywhere in free-space propagation.

These quantities do not rigorously apply to the so-calledpseudo-stigmatic beam, which is also rotationally symmetric infree-space propagation, but contains a hidden non-symmetrical parameter.This hidden parameter may not manifest itself in beam propagationthrough any rotationally symmetric optics, including the free-space, butit may appear in beam propagation through cylindrical optics. The beampropagation parameter M² remains constant in beam propagation throughrotationally-symmetric ABCD-type optical systems, and thereforerepresents a beam characteristics. It has a value which is always ≧1.Generally, the closer the beam parameter value is to 1, the better thebeam quality is considered to be.

Consider the beam parameter product, or the product between the waistsize and the divergence angle for a ST beam:D ₀θ=4M ²λ/π  [1]where λ is the wavelength of the laser light or the average wavelengthof the white light. The lower the M² value, the smaller the “beamparameter product,” and the better localized the laser beam in thenear-field (w₀ or D₀) and the far-field (θ) simultaneously.

The physical properties of a beam can be described mathematically by a4×4 real, symmetric, and positive definite beam matrix containing allthe beam second-order moments. There are a maximum of ten independentsuch moments. The relevant information on the beam propagationproperties and on the beam physical parameters is contained in this beammatrix. The 4×4 matrix is usually written as four 2×2 submatrices. Thesymmetry of these 2×2 submatrices determines the beam symmetryproperties.

A stigmatic (ST) beam is a beam which has its four 2×2 submatricesproportional to the identity matrix. Therefore a ST beam has at mostthree independent parameters. It has also a rotational symmetry in freespace. Not all rotationally symmetric beams are necessarily ST beams.

An aligned simple astigmatic (ASA) beam is a beam which has all its four2×2 submatrices diagonal. Therefore an ASA beam has at most sixindependent parameters: three for each transverse coordinate, x and y.An ASA beam has different propagation and physical parameters in the twoindependent transverse directions, x and y. An ASA beam has two beampropagation parameters, M² _(x) and M² _(y), which for some beams areequal (intrinsic stigmatic, or IS, beams), and for some other beams arenot equal (intrinsic astigmatic, or IA, beams). An ASA beam has anorthogonal symmetry of its transverse irradiance (“intensity”) profileeverywhere along its axis in free-space propagation. An ASA beam keepsthe same ASA symmetry and preserves its M² _(x) and M² _(y) factors bypropagation through aligned cylindrical optics androtationally-symmetric optics.

A general astigmatic (GA) beam is a beam which has two 2×2non-symmetrical submatrices. Some GA beams may have non-orthogonalsymmetry in their transverse irradiance profile in free-spacepropagation. Such beams are called twisted-irradiance beams, or simply,twisted beams. There are also GA beams with rotational symmetry and GAbeams with orthogonal symmetry in propagation through free-space andspherical optics, but not through cylindrical optics. Such beams aregenerally called pseudo-symmetrical beams, or more specificallypseudo-ST (PST), and pseudo-ASA (PASA) beams, respectively, ortwisted-phase beams. The maximum number of independent elements of theGA beam matrix is ten.

Any GA beam can be transformed (detwisted, or decoupled) up to an ASAbeam (with M² _(x)≠M² _(y)) or up to a ST beam (with M² _(x)=M²_(y)=M²), by using spherical and cylindrical lenses, mirrors, andsections of free-space or ABCD-type optics.

The intrinsic astigmatism parameter, a, is an invariant beam parameterwhich is a function of all ten second-order moments of a beam, whichdiscriminates between the only two existing disjoint classes of beams:intrinsic stigmatic (a=0), and intrinsic astigmatic (a>0). The intrinsicastigmatism parameter a remains constant at beam propagation throughABCD-type optics, even though the beam symmetry changes during itspropagation. The intrinsic astigmatism parameter a is a measure of thedifference between the two beam propagation parameters, M² _(x)≠M² _(y),defined for the beam transformed into its simplest, ASA-type symmetry:a=(1/2)(M ² _(x) −M ² _(y))².  [2]

An intrinsic stigmatic (IS) beam is a beam which, irrespective of itssymmetry in free space (round, orthogonal, or even non-orthogonal), isobtained from or can be transformed into a ST beam by using onlyABCD-type optics. An intrinsic astigmatic (IA) beam is a beam whichcannot be obtained from (or transformed into) a ST beam by using onlyABCD-type optics. An IA beam has a>0, and therefore cannot normally betransformed into a beam with a=0 by using the above mentioned ABCD-typeoptics. A collimated beam is a beam for which its Rayleigh range is muchgreater (usually at least 10 times greater) than all the characteristiclengths of intervening optics (free-spaces and focal lengths of thelenses).

The systems and methods described below preferably employ IS, collimatedbeams, either with ST symmetry (round shape in free space), or with ASAsymmetry (orthogonal symmetry in free space with the elliptical spotaligned to the horizontal and vertical axes). Typical examples of STbeams are the collimated beams of virtually all types of lasers,including beams from diode lasers which are passed through low-ordermode fiber and then collimated to become ST beams. Typical examples ofIS beams with ASA symmetry are the collimated beams with elliptical beamspots in free space of index-guided diode lasers without fiber optics.

The degree to which a beam is intrinsic stigmatic is approximate. Thedegree depends on the errors of measuring its parameters, and to thedegree to which we want to obtain the desired parameters, especiallyspot roundness. An IS beam can be defined as that one for which theparameter a is much less than the square of the geometrical mean (andtherefore also of the arithmetic mean) of the two beam propagationparameters, M² _(x) and M² _(y):a<<M ² _(x) M ² _(y)  [3]

Alternatively, an IS beam can be defined as a beam having the ratiobetween the two different M² _(x,y) close to one, or more precisely:1−ε<M ² _(x) /M ² _(y)<1+ε  [4]

where ε is a positive small number, ε<<1, to be conventionallyspecified. A collimated beam having in any transverse section a spotwith the aspect ratio (the ratio between the major and the minor axis ofthe transverse irradiance spot) of 1.15:1, is considered a round beameverywhere along its axis according to ISO 11146 document. This beam isapproximately an IS beam. The above criterion gives approximately ε≈0.15and alsoa≦(1/2)ε² M ² _(x) M ² _(y)≈0.01 M ² _(x) M ² _(y)  [5]

For example, using the above criteria to test whether a beam is IS orIA, a high power YAG:Nd laser with some intrinsic astigmatism, M²_(x)=40 and M² _(y)=46, can still be considered IS, having a=18<18.4,and ε=0.15; however, if the beam has slightly different values of M²,i.e. 40 and 50, then it is IA, because a=50 is not <20, and ε=0.2>0.15.Similarly, a beam from a diode laser having M² _(x)=1.5 and M² _(y)=1.7can be considered also IS, having a=0.02<0.026, and ε=0.13<0.15.Alternatively, a beam from a pulsed N₂ laser having M² _(x)=9 and M²_(y)=12 is an IA one, because a=4.5 which is not <1.08, and ε=0.33>0.15.

The description of preferred embodiments below will ordinarily employthe thin lens approximation, which neglects the physical thickness ofany lenses relative to their focal lengths. The optics (lenses, mirrors)will be considered aberration-free. Each optical element and the wholeoptical system is considered an ABCD-type system.

All the concepts and optical configurations will be illustratedspecifically for laser beams, i.e., for coherent electromagnetic waveswith wavelength in the range of 0.1 micrometers to 100 micrometers.Similar optical configurations can be implemented for coherentelectromagnetic waves with longer wavelengths, in the millimeter andcentimeter microwave range called the quasi-optical domain, by usingspecific components for the quasi-optics equivalent to the ABCD-typeoptical systems.

Preferred Embodiments

Two main beam configurations will be treated in sequence: collimated STbeams (with rotational symmetry), and collimated ASA beams (withorthogonal symmetry). Each of these beam configurations has severaldifferent embodiments, depending on the operation mode: image-mode,projection-mode, and focus-mode. For each main case, a generatingconfiguration will be described first, to allow an analysis of theproperties of this configuration, and allow an easy understanding ofsubsequent configurations and embodiments based on that generatingconfiguration.

(I) Configurations for Collimated Stigmatic (ST) Beams

1. Image Mode

FIGS. 1-A and 1-B show schematic isometric and side views, respectively,of a two-lens optical system configuration 20 according to the presentinvention. The optical axis (z-axis) is shown at 22. A first (input)cylindrical lens 24 is placed along optical axis 22 such that lens 24 isconverging along the x axis. First lens 24 is a positive-focus lens, andhas a cylindrical focal length f₁. A second (output) cylindrical lens 26is placed along optical axis 22, approximately at an optical distance2f₁ behind lens 24, as discussed below. As illustrated in FIGS. 1-A and1-B, lenses 24 and 26 are preferably separated by free space. Ingeneral, one or more lenses and/or mirrors could be placed betweenlenses 24, 26, while still allowing the positioning of second lens 26 atthe round-spot location following first lens 24. Second lens 26 can be apositive or negative focus lens, and has a cylindrical focal length f₂.The cylindrical axis of lens 26 is rotated about the z-axis with anangle α with respect to the cylindrical axis of first lens 24.

Consider an incoming collimated, ST, rotationally symmetric beamincident on first lens 24, with a waist of diameter D₀ at first lens 24.The beam has an elliptical cross-section in the space between lenses 24,26. Second lens 26 is located in the plane where there is again a roundspot after first lens 24, at or very close to a distance 2f₁ from firstlens 24. If the incoming beam is ideally collimated, second lens 26 isexactly 2f₁ away from first lens 24. If the incoming beam is slightlyconvergent (or divergent), the round spot is situated at a distanceslightly shorter (or longer) than 2f₁. Second cylindrical lens 26 isplaced in the plane of this round spot after first cylindrical lens 24.

There is always a round spot after first lens 24 if the incoming beam isa ST, rotationally symmetric, collimated beam. In classical optics thisround spot is also called the circle of least confusion for the ASA beamcreated by first lens 24. An output round spot of an adjustable diameterD(α) is generated at an output transverse plane or screen 30 located ata fixed distance d₂₀ after second lens 26, provided the followingconditions are met:f ₂>0  [6.1]f ₂ <f ₁  [6.2]d ₂₀>0.  [6.3]Conditions [6.1–3] specify, respectively, that second cylindrical lens26 is converging, second cylindrical lens 26 is more powerful than firstcylindrical lens 24, and the output round spot is after second lens 26,and real, i.e., can be seen on screen 30.

The distance d₂₀ at which the real spot is round can be called the imagedistance. The distance d₂₀ is given by:d ₂₀=2/(1/f ₂−1/f ₁)=2f ₂/(1−f ₂ /f ₁)>2f ₂  [7]The diameter of the real round spot is adjustable by changing the angleα of second cylindrical lens 26 according to the equation (exactformula):D(α)=D ₀[1+4(f ₁ /z _(R))²/(1−f ₂ /f ₁)²+(4f ₂ /f ₁)sin²(α)/(1−f ₂ /f₁)²]^(1/2)  [8.1]Eq. [8.1] can be rewritten as:D(α)=D₀ K(α),  [8.2]where K(α)>1 is the variable magnification factor, or variable scalefactor.For well collimated beams (f₁/z_(R)→0), in the geometrical opticsapproximation (marked by the index g), eq. [8] becomes:D _(g)(α)=D ₀[1+(4f ₂ /f ₁)sin²(α)/(1−f ₂ /f ₁)²]^(1/2) =D ₀ K_(g)(α).  [9]

The minimum output spot size corresponds to α=0, i.e. an orientation ofthe cylindrical axis of second cylindrical lens 26 parallel to thecylindrical axis of first lens 24. The minimum spot size D_(m)=D(0) isgiven by:D _(m) =D(0)=D ₀[(1−f ₂ /f ₁)²+4(f ₁ /z _(R))²]^(1/2)/(1−f ₂ /f₁)  [10.1]In the geometrical optics approximation, eq. [10.1] can be rewritten as:D _(mg) =D _(g)(0)=D ₀  [10.2]

The maximum spot size corresponds to α=90°, i.e. an orientation of thecylindrical axis of second cylindrical lens 26 perpendicular to thecylindrical axis of first lens 24. The maximum spot size is given by:D _(M) =D(π/2)=D ₀[(1+f ₂ /f ₁)²+4(f ₁ /z _(R))²]^(1/2)/(1−f ₂ /f₁).  [11.1]In the geometrical optical approximation the maximum spot size is:D _(Mg) =D _(g)(π/2)=D ₀(1+f ₂ /f ₁)/(1−f ₂ /f ₂)=D ₀(f ₁ +f ₂)/(f ₁ −f₂),  [11.2]which is >D₀.If we define the dynamic range factor K as the ratio of the maximum tothe minimum spot size,K=D _(M) /D _(m)  [12.1]we obtain (exact formula):K=[(1+f ₂ /f ₁)²+4(f ₁ /z _(R))²]^(1/2)/[(1−f ₂ /f ₁)²+4(f ₁ /z_(R))²]^(1/2)  [12.2]In the geometrical optics limit, eqs. [12.1–2] can be rewritten as:K _(g) =D _(Mg) /D _(mg)=(1+f ₂ /f ₁)/(1−f ₂ /f ₁).  [13]A dynamic range K of 2–20 can be readily obtained by appropriatelyselecting the values of f₁ and f₂.

By placing a transparency (object) either on the first cylindrical lens24, or on the second lens 26, in a fixed, aligned position with respectto the x and y axes, an adjustable size, proportional image is generatedat screen 30. The image at screen 30 is rotated by an angle 2α. If thetransparency/image is placed on the second cylindrical lens 26 androtated together with it, the image at screen 30 is a fixed, alignedimage, with a variable scale factor depending on α according to theequations above. For a laser beam, a certain transverse irradianceprofile at the waist on the first cylindrical lens 24 can be replicatedat screen 30 with an adjustable scale factor given by the equationsabove. The input irradiance profile can be for example a circulartop-hat (uniform, rotationally-symmetric) profile, which is ofparticular use in industrial and medical laser applications.

The configuration shown in FIGS. 1-A and 1-B is useful for replicatingand scaling a certain irradiance profile with an adjustable scale factorK(α)=2–20 at a fixed distance d₂₀, or to project some images at a fixeddistance d₂₀ with an adjustable scale factor K(α).

If in the above generating configuration the second cylindrical lens 26is either positive but weaker than the first one, f2>f1, or is negative,f2=−|f2|, the distance d₂₀ becomes negative, i.e., the beam after thesecond lens 26 is never round. However, there is a virtual round spotlocated before the second lens 26, which can be converted into a realround spot by using an additional spherical relay lens L after thesecond lens 26. The virtual round spot located at −|d₂₀| becomes avirtual object for the spherical lens L which, with appropriate focallength and positioning, can give a real image of the round spot.Overall, the adjustable virtual round spot is transformed into a realadjustable round spot. The specific values of the “virtual imagedistance”−|d₂₀|, and of the “virtual round spot size” D_(v)(α) arerelevant in this configuration, because the image location and sizeafter the relay lens L (or after any imaging system) can be determinedstraightforwardly.

For a positive but weaker second cylindrical lens, f2>f1, the distancefrom the second cylindrical lens to the virtual round spot plane locatedbefore the second lens is:d ₂₀=−2/(1/f ₁−1/f ₂)=−2f ₂/(f ₂ /f ₁−1)=−|d ₂₀|  [14]The adjustable virtual round spot size is (exact formula, where thesubscript v stands for “virtual”):D _(v)(α)=D ₀[1+4(f ₁ /z _(R))²/(f ₂ /f ₁−1)²+(4f ₂ /f ₁)sin²(α)/(f ₂ /f₁−1)²]^(1/2),  [15.1]which can be rewritten asD _(v)(α)=D ₀ K _(v)(α).  [15.2]For collimated beams, or geometric optics approximation (f₁/z_(R)→0),eq. [15.2] becomesD _(vg)(α)=D ₀[1+(4f ₂ /f ₁)sin²(α)/(f₂ /f ₁−1)²]^(1/2) =D ₀ K_(vg)(α).  [16]

The minimum virtual spot size D_(vm) is given by (for α=0, exactformula):D _(vm) =D _(v)(0)=D ₀[(f ₂ /f ₁−1)²+4(f ₁ /z _(R))²]^(1/2)/(f ₂ /f₁−1)  [17.1]In the geometric optics approximation the minimum virtual spot is:D _(vmg) =D _(vg)(0)=D ₀  [17.2]The maximum spot size is given by (for α=90°, exact formula):D _(vM) =D _(v)(π/2)=D ₀[(f ₂ /f ₁+1)²+4(f₁ /z _(R))²]^(1/2)/(f ₂ /f₁−1)  [17.3]In geometrical optics approximation the maximum spot size is:D _(vMg) =D _(vg)(π/2)=D ₀(f ₂ /f ₁+1)/(f ₂ /f ₁−1)=D ₀(f ₂ +f ₁)/(f ₂−f ₁)  [17.4]The dynamic range factor K_(v) (exact) or K_(vg) (geometrical optics)is:K _(v)=[(f ₂ f ₁+1)²+4(f ₁ /z _(R))²]^(1/2)/[(f ₂ /f ₁−1)²+4(f ₁ /z_(R))²]^(1/2)  [17.5]K _(vg)=(f ₂ /f ₁+1)/(f ₂ /f ₁−1)=(f₂ +f ₁)/(f ₂ −f ₁)  [17.6]

For a negative second cylindrical lens 26, f₂=−|f₁|<0, the focal lengthof the second cylindrical lens 26 is unrestricted by the focal length ofthe first cylindrical lens 24. The distance from the second cylindricallens 26 to the virtual round spot plane located before second lens 26is:d ₂₀=−2/(1/f ₁+1/|f ₂|)=−2|f ₂|/(1+|f ₂ |/f ₁)=−|d ₂₀|  [18.1]The adjustable virtual round spot size D_(v)(α) is (exact formula):D _(v)(α)=D ₀[1+4(f ₁ /z _(R))²/(1+|f ₂ |/f ₁)²−(4|f ₂ |/f₁)sin²(α)/(1+|f ₂ |/f ₁)²]^(1/2)  [18.2]which becomes for collimated beams, or geometric optics approximation(f₁/z_(R)→0):D _(vg)(α)=D ₀[1−(4|f ₂ |/f ₁)sin²(α)/(1+f ₂ |/f ₁)²]^(1/2)  [18.3]Equations [18.2–3] can be rewritten as:d _(v)(α)=D ₀ K _(v)(α),D _(vg)(α)=D ₀ K _(vg)(α)  [18.4]The minimum virtual spot size D_(vm) is obtained for α=90° and is givenby:D _(vm) =D _(v)(π/2)=D ₀[(1−|f ₂ |/f ₁)²+4(f ₁ /z _(R))²]^(1/2)/(1+|f ₂|/f1)  [18.5]In geometric optics approximation it is:D _(vmg) =D _(vg)(π/2)=D ₀|1−|f ₂ /f ₁/(1+|f ₂ |/f ₁)=D ₀ |f ₁ −|f₂||/(f ₁ +|f ₂|)  [18.6]The maximum virtual spot size is given by (for α=0):D _(vM) =D _(v)(0)=D ₀[(1^(+|f) ₂ |/f ₁)²+4(f ₁ /z _(R))²]^(1/2)/(1+f ₂|/f ₁)  [18.7]In geometric optics approximation it is:D _(vMg) =D _(vg)(0)=D ₀  [18.8]The dynamic range factor K_(v) (exact formula) and its counterpartK_(vg) in geometrical optics are:K _(v)=[(1+|f ₂ |/f ₁)²+4(f ₁ /z _(R))²]^(1/2)/[(1−|f ₂ |/f1)²+4(f ₁ /z_(R))²]^(1/2)  [18.9]K _(vg)=(1+|f ₂ |/f ₁)/|1−|f ₂ |/f ₁|=(f ₁ |f ₂|) /|f ₁ −|f ₂||  [18.10]

While it is possible to use a relay spherical lens also for a real roundspot, in the configuration of FIGS. 1-A–B, for practical applicationsany of the configurations with a virtual round spot can be moreadvantageous because they allow a relatively shorter overall length ofthe total configuration (two cylindrical lenses and a spherical lens).

A negative first cylindrical lens can be used in conjunction with apositive or negative second cylindrical lens, by positioning anappropriate spherical lens between the two cylindrical lenses such thatthe second cylindrical lens is situated in the round-spot planefollowing the first spherical lens.

2. Projection Mode

In the configuration of FIGS. 1-A and 1-B, the image distance d₂₀ can bechosen to be infinity, d₂₀→∞, by having both cylindrical lenses 24, 26positive and of equal focal length, f. A real round spot of adjustablesize is then located far away from the second cylindrical lens 26. Thatspot has all the properties of the real round and adjustable spot from afinite image distance. Therefore, a variable size image can be projectedon a screen located far away from the second cylindrical lens 26.

Consider the optical system of FIGS. 1-A–B, with f₁=f₂=f>0. FIGS. 2-Aand 2-B show a schematic geometric optics illustration of the passage ofa light beam through system 20, for two extreme angular positions ofsecond lens 26, respectively. FIG. 2-A corresponds to parallelcylindrical axes for the two lenses 24, 26, while FIG. 2-B correspondsto mutually perpendicular cylindrical axes.

An incoming, perfectly collimated round beam has a diameter D₀ andsubstantially zero divergence. The beam divergence, D₀/z_(R), is closeto zero for a well-collimated beam. In the parallel-orientationconfiguration shown in FIG. 2-A, first lens 24 focuses thex-distribution 40 a of the incoming beam at a distance f, and leaves they-distribution 40 b of the incoming beam unaltered. Second lens 26recollimates the x-distribution 40 a, and leaves unchanged the ydistribution of the beam (with dotted line in FIG. 2). After passagethrough second lens 26, the output light beam is collimated.

In the perpendicular-orientation illustrated in FIG. 2-B, the secondcylindrical lens 26 is rotated perpendicular to the first cylindricallens 24. First lens 24 acts only on the x-distribution 44 a of the lightbeam, while second lens 26 acts only on the y-distribution 44 b of thelight beam. The x-distribution 44 a of the beam is focused by the firstcylindrical lens 24, and remains divergent after the second cylindricallens 26. The y-distribution 44 b of the beam is collimated until thesecond cylindrical lens 26, and then leaves the second cylindrical lensas a converging, and then diverging y-distribution. At a distance farenough from the second cylindrical lens 26, both x-distribution 44 a andy-distribution 44 b have the same maximum divergence, which is, ingeometrical optics approximation, θ_(M)=D₀/f. By rotating the secondcylindrical lens 26, a variable divergence can be varied between aminimum value (D₀/z_(R), which is near zero,) and a maximum value(D₀/f). A rigorous result can be obtained from Eqs. [7] and [8] bydefining the variable divergence θ(α) as the limit (for d₂₀→∞, or forf₂→f₁=f):

$\begin{matrix}{d_{20} = {{\lim\limits_{{{f2}\rightarrow{f1}} = f}\left\lbrack {2/\left( {{1/f_{2}} - {1/f_{1}}} \right)} \right\rbrack} = \infty}} & \lbrack 19.1\rbrack\end{matrix}$

$\begin{matrix}{{\theta(\alpha)} = {{\underset{{{f2}\rightarrow{f1}} = f}{\lim\limits_{{d_{20}\rightarrow\infty},{or}}}{{D(\alpha)}/d_{20}}} = {\left( {D_{0}/f} \right)\left\lbrack {\left( {f/z_{R}} \right)^{2} + {\sin^{2}(\alpha)}} \right\rbrack}^{\frac{1}{2}}}} & \lbrack 19.2\rbrack\end{matrix}$In geometrical optics approximation the following variable divergence θg(α) results:

$\begin{matrix}{{\theta_{g}(\alpha)} = {{\underset{{{f2}\rightarrow{f1}} = f}{\lim\limits_{{d_{20}\rightarrow\infty},{or}}}{{D_{g}(\alpha)}/d_{20}}} = {\left( {D_{0}/f} \right){\sin(\alpha)}}}} & \lbrack 19.3\rbrack\end{matrix}$which gives correct results for α16 0. For α=0, the exact formula belowgives the correct result of the minimum divergence:θ_(m)=θ(0)=D ₀ /z _(R)=θ_(in),  [19.4]where θ_(in) is the initial divergence of the incoming beam.Note that in the geometrical optics approximation we have thenon-physical result:θ_(mg)(0)=θ_(g)(0)=0.  [19.5]The maximum divergence is obtained for α=90° and is given by (the exactformula and the geometrical optics, respectively):θ_(M)=θ(π/2)=(D ₀ /f)[1+(f/z _(R))²]^(1/2)  [19.6]θ_(Mg)=θ_(g)(π/2)=D ₀ /f  [19.7]The dynamic range factor K, defined as K=θ_(M)/θ_(m) (exact formula) andits geometrical optics counterpart K_(g) are given by:K=[1+(z _(R) /f)²]^(1/2)  [19.8]

$\begin{matrix}{K_{g} = {{\lim\limits_{z_{R}\operatorname{>>}f}K} = {z_{R}/f}}} & \lbrack 19.9\rbrack\end{matrix}$

Desired values for the ratio K or K_(g) in Eqs. [19.6–9] between themaximum and minimum image size at a certain distance d can be obtainedby using the optical system of FIGS. 1-A–B to project an image withvariable magnification as described above. The maximum, D_(M), and theminimum, D_(m), image size at the distance d, such as f<<d<<z_(R), aregiven by multiplying the maximum and the minimum divergence with thedistance d:D _(M)=θ_(Mg) ^(d=D) ₀ d/f  [19.10]D _(m)=θ_(m) d=D ₀ d/z _(R)  [19.11]

Consider now a projection-mode optical system with positive and negativecylindrical lenses of equal absolute power, f₂=−f₁=−f. FIG. 3 shows anisometric view of a system 120 having two such lenses 124, 126 placed asclose to each other as mechanically possible. The distance betweenlenses 124, 126 is at least an order of magnitude (factor of 10) lowerthan the focal length f. The optical axis of system 120 is illustratedat 122, while the beam cross-section at the two lenses is illustrated at132. Either one or both of the two lenses can be rotated with respect toeach other as described above. The system of FIG. 3 is functionallysimilar to a projection-mode system implemented in the configurationshown in FIGS. 1-A–B. The system of FIG. 3 can act either as a variabledivergence optical system for a laser beam, or as a variablemagnification projection system. In such a variable-magnificationprojection system, a collimated white light beam can be used at itsinput, and a transparent object can be attached to the rotating lens.The target projection screen is preferably positioned at a distance atleast ten times larger than the magnitude of the focal length f.Ideally, in the system of FIG. 3, the two cylindrical lenses 124, 126are infinitely thin and superimposed at the same physical location. Inpractice, cylindrical lenses such as lenses 124, 126 have finitethicknesses. As a result, lenses 124, 126 may not cancel each other whenaligned.

FIG. 4 shows a schematic side view of an optical system 220 which imagesa focal plane 246 of a positive input cylindrical lens 224 to a focalplane 248 of a negative output cylindrical lens 226. The optical axis ofsystem 220 is shown at 222. Two identical, positive-magnificationspherical lenses 250 a–b of focal length f₀ are positioned betweencylindrical lenses 224, 226 in a −1 magnification afocal configuration.Lenses 250 a–b are separated by a distance 2 f₀. The first principalplane 246 of the first cylindrical lens 224 coincides with the frontfocal plane of a first spherical lens 250 a. The first principal plane248 of the second cylindrical lens 226 coincides with the back focalplane of the second spherical lens 250 b. The total physical length ofthe configuration is approximately 4f₀. The first principal planes oflenses 224, 226 coincide. Similarly, the second principal planes oflenses 224, 226 coincide.

3. Focus Mode

FIG. 5 shows a schematic isometric view of a focus-mode optical system320 according to another embodiment of the present invention. Theoptical axis of system 320 is illustrated at 322. System 320 includesfirst and second cylindrical lenses 324, 326 of focal length f, and aspherical lens 360 of focal length f₀, situated after second lens 326.At least one of lenses 324, 326 is capable of rotation about the opticalaxis 322. An output plane or screen 330 is situated in the back focalplane of spherical lens 360, at a distance f₀ away from spherical lens360. The working distance of system 320 is equal to f₀. Cylindricallenses 324, 326 are separated by an optical distance 2f, as describedabove with reference to FIGS. 1-A–B. Spherical lens 360 is situated adistance s≈0 after second cylindrical lens 326. Spherical lens 360 canbe attached to the second, rotating cylindrical lens 326 in order toreduce the total length of system 320.

Irrespective of the placement of a focusing spherical lens in anincoming beam, the spot size at its back focal plane, D_(f), isproportional to the divergence of the incoming beam, θ_(in):D _(f)=θ_(in) f ₀,  [20.1]where f₀ is the focal length of the convergent spherical lens. Thevariable spot size D_(f)(α) in the focal plane of spherical lens 360 isgiven by the (exact) formula:D _(f)(α)=(D ₀ f ₀ /f)[(f/z _(R))²+sin²(α)]^(1/2) =D ₀ K(α)  [20.2]In a geometrical optics approximation the variable spot size D_(fg)(α)is given by:D _(fg)(α)=D ₀(f ₀ /f)sin(α)=D ₀ K _(g)(α)  [20.3]The minimum spot size is obtained for a=0 and is given by:D _(fm) =D _(f)(0)=D ₀ f ₀ /z _(R)=θ_(in) f ₀  [20.4]

The minimum spot size of Eq. [20.4] is the minimum spot size which wouldbe obtained in the focal plane of the same spherical lens of focallength f₀, without any intervening cylindrical optics, for an identicalincoming beam having an incoming divergence θ_(in). In pure geometricaloptics the following non-physical result is obtained:D _(fmg) =D _(fg)(0)=0  [20.5]The maximum spot size D_(fm) corresponds to α=90° and is given by theexact relations:D _(fm) =D _(f)(π/2)=(D ₀ f ₀ /z _(R))[1+(z _(R) /f)²]^(1/2)=(D ₀ f ₀/f) [1+(f/z _(R))²]^(1/2)  [20.6]D _(fM)=θ_(in) f ₀[1+(z _(R) /f)²]^(1/2)  [20.7]For a well collimated beam (geometrical optics approximation) themaximum spot size is:D _(fMg) =D _(fg)(π/2)=D ₀ f ₀ /f  [20.8]The exact dynamic range factor K and the geometrical optics dynamicrange factor Kg are given respectively by:K=[1+(z _(R) /f)²]^(1/2)  [20.9]

$\begin{matrix}{K_{g} = {{\lim\limits_{z_{R}\operatorname{>>}f}K} = {z_{R}/f}}} & \lbrack 20.10\rbrack\end{matrix}$

A remarkable feature of the “focus-mode” arrangement shown in FIG. 5 isits large achievable dynamic range factor. Dynamic range factor valuesof 50 to 400 are readily achievable. The design of FIG. 5 is alsoremarkably flexible. The desired working distance determines the focallength of spherical lens 360. The desired dynamic range factordetermines the required incoming beam collimation (the value of itsRayleigh range z_(R)) and the cylindrical lens characteristics. For alower value of K and a large Rayleigh range, the positive-negativecylindrical lens configuration of FIGS. 3–4 can be more convenient thanthe positive—positive configuration of FIGS. 1-A–B. The depth of focus,or the extent over which the beam maintains approximately a round spot,can be a relevant parameter in certain applications. An estimate forthis range is by considering the true Rayleigh range of the focused beamwhen the minimum spot is obtained, z_(RDm), given by:Z _(RDm) =f ₀ ² /z _(R)  [20.11]One can estimate a deviation from the round spot within this“quasi-Rayleigh range” of about 10%, for the entire range of spot sizesat the working distance.

Focus-mode optical systems such as the system 320 shown in FIG. 5 can beparticularly useful in beam delivery systems for industrial and medicallasers. The input beam can be collimated accordingly before entry intosystem 320, either using discrete lenses and free-spaces, or usingoptical fibers and collimating lenses. System 320 allows the use of asingle laser for a relatively wide range of applications, by allowingrelatively wide variations in the laser spot size at the target and thusallowing adjustments of the laser power density or the laser energydensity at the target plane. Applications such as cutting and surfacetreatment for industrial lasers, and medical applications in dermatologyand ophthalmology would benefit from variable spot size optical systemssuch as the ones described above.

A focus-mode system can be constructed by modifying the configurationshown in FIG. 3, by positioning a convergent spherical lens adjacent tothe second cylindrical lens 126 opposite the first cylindrical lens 124.The target location where a symmetrically-scalable round spot is formedis then situated a distance f₀ away from the spherical lens, where f₀ isthe focal length of the spherical lens. Several such compactconfigurations employing adjacent lenses are described below withreference to FIGS. 3, 9 and 10.

(II) Collimated, Aligned Simple Astigmatic (ASA) Beams

Collimated ASA beams of intrinsic stigmatic type can be obtained bycollimating an index-guided diode laser. The extent of such a beam overone of the transverse axes, for example the x-axis, is several timeslarger than the extent of the beam over the other transverse axis. Abeam aspect ratio of 3:1 to 5:1 is typical.

There are in principle two ways of generating a variable-size round spotat a desired working distance (say, at infinity) using a laser sourcewith a collimated elliptical beam. One way is to enlarge the small axisof the elliptical beam until it equals the large axis, and then totransform the resulting round, collimated laser beam into avariable-divergence beam. The second way is to reduce the large axis ofthe elliptical beam to make it equal to the small axis, and subsequentlyto transform the resulting round, collimated beam into avariable-divergence beam. Assume for example that we are interested inenlarging the small axis of the beam until it becomes as large as thelarger axis. The enlargement could be achieved by using anegative-positive pair of cylindrical lenses, or a positive—positivepair. Reducing the larger axis of the beam can be achieved by switchingthe direction of light travel through the lens pair used for enlargement(i.e. switching the system input and output). Therefore, there are fourdifferent configurations that could be used to generate a round spot ata desired distance using an incoming ASA beam: small-to-largeconfiguration using negative and positive cylindrical lenses (Galileanmagnifying telescope with the positive lens rotating); small-to-largeconfiguration using positive and positive cylindrical lenses (Keplerianmagnifying telescope with rotating second lens); large-to-smallconfiguration using positive and negative cylindrical lenses (Galileandemagnifying telescope with rotating negative lens); and large-to-smallconfiguration using positive and positive cylindrical lenses (Kepleriandemagnifying telescope with rotating second lens). Only one of the fourpossible configurations will be analyzed in detail below. The otherthree configurations will be readily apparent to the skilled artisanfrom the discussion below.

FIGS. 6-A–B show schematic views of a small-to-large cylindricaltelescope 420 having a first, fixed, negative focus cylindrical lens424, and a second, rotatable, positive focus cylindrical lens 426.Although the discussion below focuses on a projection-mode, variabledivergence system, corresponding focus-mode systems can be constructedby adding an output spherical lens as described above. First lens 424has a focal length f₁=−|f₁|, while second lens 426 has a positive focallength f₂. First lens 424 is oriented so as to act only along thesmaller beam size direction, or the y-direction. Second lens 426 issituated at the location where the light beam is round after first lens424. This position corresponds approximately to a distance f₂−f₁ afterfirst lens 424.

The incoming beam is collimated on both the x- and y-directions, and hasboth its x- and y-waists at first lens 424. The beam has a larger waistsize along the x-axis, D_(0x), than along the y-axis, D_(0y). Whensecond lens 426 is parallel with first lens 424, i.e. when second lens426 acts also only along the y-axis, the configuration is a Galileanmagnifying telescope (or afocal system), as illustrated in FIG. 6-A. Thex-distribution 440 a of the beam passes undisturbed through bothcylindrical lenses 424, 426 and retains its size D_(0x). They-distribution 440 b is enlarged by f₂/|f₁|>1 by the Galilean afocalsystem. To obtain a collimated round spot after second lens 426, themagnification ratio of the telescope should equal the ratio of the beamsizes along the x- and y-directions:D _(0x) /D _(0y) =f ₂ /|f ₁|  [21.1]The divergence of the beam with both lenses in parallel position actingon y is minimum (zero in geometric optics limit), or equal to thesmaller divergence of the larger axis, x:θ_(m)=θ_(x) =D _(0x) /z _(Rx)  [21.2]FIG. 6-B illustrates the effect of rotating second cylindrical lens 426by 90°. As shown, both the x- and y-distributions 444 a–b of the lightbeam become divergent, with the maximum divergence angle θ_(M) given by:θ_(M) =D _(0x) /f ₂ =D _(0y) /|f ₁|  [21.3]The cylindrical Galilean afocal optical system 420 with the rotatingsecond lens 426 is a variable divergence source of light for acollimated ASA beam of intrinsic stigmatic type. The other threepossible configurations can be analyzed similarly.

As the skilled artisan will appreciate from the preceding discussion, itis possible to construct projection-mode optical systems with variablespot sizes at large distances for collimated ASA beams of intrinsicstigmatic type. Adding a spherical lens after the second cylindricallens allows generating focus-mode, variable spot sizes at a fixedworking distance. Applications such as coupling to a fiber, materialprocessing, or medical applications (dermatology) can benefit from thedesigns described above.

(III) Systems Using Cylindrical Mirrors

Generally, the lenses described above can be replaced by suitably-chosenmirrors. In the far infrared spectral domain, reflective optics aresometimes less expensive than refractive optics. Mirrors can also be ofparticular use when the chromatic aberrations are of concern. Ingeneral, many configurations using mirrors or mirrors and lenses can beconceived.

FIG. 7 shows a configuration 520 using a first (input) cylindrical lens524, and a second (output) cylindrical mirror 526 situated along theoptical axis 522 of the incoming beam. Lens 524 is a converging lens offocal length f, acting along the y-axis. Mirror 526 is a converging,rotatable cylindrical mirror with a focal length f and a cylindricalradius of curvature 2f. A polarizing cube beam splitter (PCBS) 550 andan attached quarter-wave plate (QWP) 552 are positioned in the opticalpath between lens 524 and mirror 526.

An input ST, collimated beam, linearly polarized along the verticaldirection, is incident on lens 524. The optical distance between thecylindrical lens 524 and the cylindrical mirror 526 is 2f. Thus, mirror526 is positioned in the round spot of the light beam after its passagethrough PCBS 550. The incoming beam is p-polarized inside PCBS 550 andis transmitted almost completely toward mirror 526. The beam reflectedfrom mirror 526 becomes s-polarized inside PCBS 550 by passing a secondtime trough QWP 552, and is reflected almost completely downwards withinPCBS 550. With mirror 526 oriented parallel to lens 524 (with bothacting along the y-direction), the reflected beam after mirror 526 has aminimum divergence equal to that of the incoming beam. With mirror 526rotated by 90° about the optical axis 522, the outgoing beam has amaximum divergence. The configuration shown in FIGS. 7 otherwisesatisfies the equations set forth above for the system shown in FIGS.1-A–B, with f₁=f₂=f.

FIG. 8 shows a configuration 620 including a first cylindrical mirror624 and a second cylindrical mirror 626 situated along an optical axis622. First mirror 624 has focusing power along the y-(vertical) axis,and has a focal length f and a cylindrical radius of curvature 2f.Second mirror 626 is rotatable about the vertical axis of FIG. 8, andhas the same focal length f and cylindrical radius of curvature 2f. Apolarizing cube beam splitter (PCBS) 650 and two attached quarter-waveplates (QWP) 652, 654 are positioned in the optical path between mirrors624, 626. The optical distance between each mirror 624, 626 and thecenter of PCBS 650 is f.

A collimated ST beam, linearly polarized along the vertical direction,is incident on PCBS 650 along optical axis 622. The beam axis paththrough PCBS 650 is established by the polarization state of the beam.The incoming beam is p-polarized and is transmitted almost completely tofirst mirror 624 through the first QWP 652. After reflection by mirror624, the light beam passes again through first QWP 652 and becomess-polarized. The s-polarized beam is reflected upwards by PCBS 650,through second QWP 654 and toward second mirror 626. The beam reflectedby second mirror 626 arrives inside PCBS 650 after a second pass throughsecond QWP 654. The beam, now p-polarized inside PCBS 650, istransmitted almost completely downward toward the system output.

The divergence of the output beam is minimal and equal to the divergenceof the incoming collimated beam for the mirrors in the position asdrawn. The maximum divergence of the output beam is obtained for thesecond mirror 626 rotated with 90° about its optical axis in FIG. 8.Second mirror 626 is located at an optical distance equal to 2f fromfirst mirror 624, where the round spot after first mirror 624 islocated. Thus the incident beam will be affected by the pair of mirrors624, 626 according to the equations described above for theconfiguration of FIGS. 1-A–B, with f₁=f₂=f.

(IV) Additional Compact Configurations Using Adjacent Lenses

As described above with reference to FIG. 5, a focus-mode system can beconstructed by modifying the configuration shown in FIG. 3, bypositioning a convergent spherical lens (shown in FIG. 5) adjacent tothe pair of cylindrical lenses shown in FIG. 3. The spherical lens ispositioned as close as mechanically feasible to the pair of adjacentcylindrical lenses, at a distance much smaller (e.g. at least a factorof 10 smaller) than the focal length of the spherical lens. A systemresulting from a modification as described above is shown schematicallyin FIG. 9.

FIG. 9 illustrates an optical system 820 including positive and negativecylindrical lenses 824, 826, respectively, each or both rotatable aboutan optical axis 822, and a converging spherical lens 860. All threelenses 824, 826, 830 are thin and positioned as close as mechanicallypossible, such that each lens acts on the same waist plane of anincoming stigmatic, round, and collimated beam. The waist plane is shownas a spot 832. System 820 works in a similar manner irrespective of theorder of the lenses. For convenience, the description below focuses onthe lens order shown in FIG. 9. In FIG. 9, the first lens 824 is apositive cylindrical lens, with a focal length f₁=f>0. The second lens826 is a negative cylindrical lens with a focal length f₂=−f₁=−f<0. Thethird lens 830 is a spherical lens with a focal length f₀>0. System 820acts as a focus-mode system, by providing a round spot 836 with anadjustable diameter, at a target 830. The target location, where roundspot 836 is obtained, is at a distance d₂₀ from the incoming waistplane, in the focal plane of the converging spherical lens:d ₂₀ =f ₀>0  [22]

By changing the relative angle α between the cylindrical axes of thecylindrical lenses, the diameter D_(f)(α) of round spot 836 on target830 is given by:D _(f)(α)=D ₀[(f ₀ /z _(R))²+(f ₀ /f)²sin²(α)]^(1/2)  [23.1]where D₀ is the diameter of the incoming waist, and subscript f standsfor the focal plane of the spherical lens. The same diameter is given,in the geometrical optics approximation, for a well collimated incomingbeam, z_(R)→∞, by:D _(fg)(α)=D ₀(f ₀ /f)|sin(α)|.  [23.2]The minimum spot size D_(fm) on the target is obtained for α=0, and is:D _(fm) =D _(f)(0)=D ₀ f ₀ /z _(R)  [23.3]and in the geometrical optics limit gives the non-physical resultD_(fm)=0. Note that the minimum spot size is determined only by theincoming beam parameters (D₀ and z_(R)) and the spherical lens (f₀), theminimum spot being diffraction-limited for ideal thin lenses. Themaximum spot size on target, D_(fm) is:D _(fm) =D _(f)(π/2)=D ₀[(f ₀ /z _(R))²+(f ₀ /f)²]^(1/2)  [23.4]In geometric optics approximation the maximum spot size is:D _(fMg) =D ₀ f ₀ /f  [23.5]

The dynamic range factor K=D_(fm)/D_(fm) is given by:K=[1+(z _(R) /f)²]^(1/2)  [23.6]and is therefore determined only by the parameters of the incoming beam(z_(R)) and the power of the cylindrical lenses (f). In the geometricaloptics limit, K_(g)→∞, which is a consequence of the non-physical zerominimum spot size diameter. In practice, system 820 allows a largedynamic range, on the order of 10–400, and therefore a substantiallybroad range of potential effects to be obtained with the same laser andoptical system. Such a system is useful for industrial, medical, andscientific applications of lasers where such a large dynamic range isdesired.

Several other configurations are described below with reference to FIGS.10 and 3. Two configurations including two cylindrical lenses of thesame sign are described with reference to FIG. 10, and twoconfigurations including two cylindrical lenses of opposite sign aredescribed with reference to FIG. 3.

FIG. 10 shows an isometric view of a system 720 including two adjacentcylindrical lenses 724, 726. Lenses 724, 726 are preferably bothpositive. The optical axis of the system 720 is illustrated as 722.Lenses 724, 726 are adjacent to each other, such that both lenses 724,726 act on the same round spot 732 of an incoming stigmatic, collimatedbeam. The separation between lenses 724, 726 is much smaller (e.g. atleast ten times smaller) than the focal length of each of the twolenses. Preferably, the separation between lenses 724, 726 is as smallas mechanically possible while still allowing relative rotation of thetwo lenses, for example less than about 1–2 mm. In some embodiments, theseparation between lenses 724, 726 may be at least 100 times smallerthan the focal lengths of the two lenses. Either one or both of the twolenses 724, 726 can be rotated with respect to each other about theoptical axis 722. The system of FIG. 9 for convergent cylindrical lenseswith different powers, f₁>0, f₂>0, f₁≠f₂ can be thought to correspond tothe image-mode system implemented with the configuration shown in FIG.1-A. A real, symmetrically-scalable round spot 736 is generated on atarget screen 730, provided screen 730 is placed at an appropriatedistance d₂₀ given byd ₂₀=2/(1/f ₁+1/f ₂)>0.  [24]

The round spot 736 represents the image of the incoming round spotcorresponding to the incoming beam waist (spot) 732 of diameter D₀. If atransparent object or mask is attached rigidly to a rotating lens (e.g.lens 726) then a symmetrically-scaled version of the object in a fixed,non-rotating position appears on screen 730 in the image spot 736. Thediameter D(α) of the adjustable round spot 736, as a function of therelative rotation angle α, is given by:D(α)=D ₀[1+4(f ₂ /z _(R))²/(1+f ₂ /f ₁)²−4(f ₂ /f ₁)sin²(α)/(1+f ₂ /f₁)²]^(1/2)  [25.1]In the geometric optics limit, where z_(R)→∞, equation [25.1] simplifiesto:D _(g)(α)=D ₀[1−4(f ₂ /f ₁)sin²(α)/(1+f ₂ /f ₁)²]^(1/2)  [25.2]

The minimum spot diameter, D_(m), is obtained when the two lenses areperpendicular, α=90°, and is given by:D _(m) =D(π/2)=D ₀[(1−f ₂ /f ₁)²+4(f ₂ /z _(R))²]^(1/2)/(1+f ₂ /f₁)  [25.3]In geometrical optics approximation this is D_(mg), given by:D _(mg) =D ₀ |f ₁ −f ₂|/(f ₁ +f ₂)  [25.4]The maximum spot diameter D_(M) is obtained for parallel cylindricallenses, α=0, and is given by:D _(M) =D(0)=D ₀[1+4([f ₂ /z _(R))²/(1+f ₂ /f ₁)²]^(1/2)  [25.5]In geometrical optics limit it becomes D_(Mg), given by:D _(Mg) =D ₀  [25.6]The dynamic range factor K=D_(M)/D_(m) and its expression in geometricaloptics limit, K_(g), are:K=[(1+f ₂ /f ₁)²+4(f₂ /z _(R))²]^(1/2)/[(1−f ₂ /f ₁)²+4(f ₂ /z_(R))²]^(1/2)  [25.7]K _(g)=(f ₁ +f ₂)/|f ₁ −f ₂|  [25.8]

Another particularly useful configuration using two positive cylindricallenses is derived from the configuration described above by usingidentical converging cylindrical lenses, i.e. for f₁=f₂=f>0. Such aconfiguration is a focus-mode configuration corresponding to thatdescribed with reference to FIG. 5. Design equations corresponding toEqs. [24] and [25.1]–[25.8] can be obtained directly from Eqs. [24] and[25.1]–[25.8] with f₁=f₂=f, and are not reproduced for simplicity. Thefocus-mode behavior results from the fact that the target plane issituated at a distance d₂₀=f.

In some embodiments, lenses 724, 726 may both be negative. In suchembodiments, the pair of cylindrical lenses may be placed on aconvergent section of the beam of interest.

Two other configurations can be understood with reference to FIG. 3. Asdescribed above, FIG. 3 shows an isometric view of a system 120 havingtwo cylindrical lenses 124, 126 placed as close to each other asmechanically feasible. The optical axis of system 120 is illustrated at122. Both lenses 124, 126 act on the same round spot 132 representingthe waist of an incoming stigmatic, circular, and collimated light beam.One or both of the lenses can be rotated about the optical axis, and theadjustable relative angle between their cylindrical axes is α.

It can be shown that irrespective of the order of the lenses (the firstlens can be the positive one, as in FIG. 3, or the negative one), system120 provides a real round spot at a positive distance after the secondlens, and therefore acts as an image-mode system, provided the negativelens is weaker in absolute value than the positive lens, i.e. f₁>|f₂|,where f₁>0 is the focal length of the first, converging cylindrical lensof FIG. 3, and f₂<0 is the focal power of the second, diverging lens ofFIG. 3.

Consider the configuration of FIG. 3, with a rotatable second lens, anda transparent object or mask rigidly attached to the rotating lens suchthat it rotates with the lens. The image of the mask appearssymmetrically-scaled with α-dependent scale factors, in a fixed,non-rotating position, at a target located at the distance d₂₀>0 afterthe second cylindrical lens, given by:d ₂₀=2/(1/f ₁−1/|f ₂|)>0  [26]

The angle-dependent diameter D(α) of a round spot at the target plane isgiven by:D(α)=D ₀[1+4(|f ₂ |/z _(R))²/(|f ₂ |f ₁−1)²+4(|f ₂ |/f ₁)sin²(α)/(|f ₂|f ₁−1)²]^(1/2)  [27.1]where D₀ is the diameter of the incoming beam waist on both cylindricallenses. For geometrical optics (subscript g) corresponding to a wellcollimated incoming beam, z_(R)→∞, the above equation becomes:D _(g)(α)=D ₀[1+4(|f ₂ |/f)sin²(α)/(|f ₂ |/f ₁−1)²]^(1/2)  [27.2]The minimum round spot diameter D_(m) at the image plane, obtained forα=0 (parallel lenses), is given by:D _(m) =D(0)=D ₀[1+4(|f ₂ |/z _(R))²/(|f ₂ |/f ₁−1)²]^(1/2)  [27.3]For geometrical optics limit this minimum spot becomes:D _(mg) =D ₀  [27.4]The maximum round spot diameter, D_(M), is obtained at the target planefor the cylindrical lenses in perpendicular position relative to eachother, α=90°:D _(M) =D(π/2)=D ₀[(|f ₂ |/f ₁+1)²+4(|f ₂ |/z _(R))²]^(1/2) /||f ₂ |/f₁−1|  [27.5]In geometrical optics approximation the maximum spot diameter is:D _(Mg) =D ₀(|f ₂ |+f ₁)/||f ₂ |−f ₁|  [27.6]The dynamic range factor K=D_(M)/D_(m) and its geometrical optics valueK_(g) are:K=[(|f ₂ |/f ₁+1)²⁺⁴(|f ₂ |/z _(R))²]^(1/2)/[(|f ₂ |/f ₁−1)²⁺⁴(|f ₂ |/z_(R))²]^(1/2)  [27.7]K _(g)=(|f ₂ |+f ₁)/||f₂ |−f ₁|  [27.8]

Such a system and the other previously-described image-mode systems areuseful when a laser beam with a specific irradiance profile (e.g. aflat-top beam) is to be replicated in the image plane, and when anon-uniform image is to be projected with a variable-scale factor on thetarget located in the image plane. Scale factors (dynamic ranges K) inthe range of 2–20 are readily achievable with the systems describedabove.

Another image-mode system can be generated by using a similar pair ofindependently rotatable positive/negative cylindrical lenses, like thelenses 124 and 126 illustrated in FIG. 3, with the negative lensstronger in absolute value than the positive lens, i.e., |f₂|<f₁, wheref₁>0 is the focal length of the first, positive cylindrical lens, andf_(2 =−|f)2 |<0 is the focal length of the second, negative cylindricallens. In this case there is no round spot at any positive distance afterthe second lens, but there is a virtual round spot, before the pair ofcylindrical lenses. In other words, the distance from the cylindricallens pair to the image plane is negative, d₂₀=−|d₂₀|<0. This virtualround spot can be made real by using a relay spherical lens after thesecond cylindrical lens, to image the virtual round spot (a virtualobject before the focal plane of the spherical lens) into a real roundspot (a real image, after the focal plane of the spherical lens). Such asystem can be thought of as a compact version of any of the two systemsdescribed by Eqs. [14]–[18]. The placement and the focal length of thespherical lens and the magnification of the imaging system using thespherical lens can be calculated using an object-image formula. Thedistance from the waist plane (or the plane of both thin cylindricallenses) to the virtual round spot, d₂₀, is given by:d ₂₀ =−d ₂₀=−2/(1/|f ₂|1/f ₁)<0  [28]Equations [27.1]–[27.8], which describe the variable virtual spot size,the minimum and maximum virtual spot size, and the dynamic factor K,apply for this configuration both as exact formulae and in the geometricoptics approximation.(V) Mechanical, Electrical and Computer Elements

The distances between the cylindrical lenses/mirrors affect theperformances of all configurations, and are also slightly dependent onthe input beam characteristics. Ideally, an optical system as describedabove is designed so as to allow a fine adjustment of these distancesafter the lenses/mirrors are assembled. Because this adjustment is doneonly once, during the assembly of the real device, expensive finemechanisms can generally be avoided. A translational adjustment allowinga range of +/−3% of the nominal length would be of particular use. Afterthe adjustment, a simple locking system (screws or glue) can be used tofix the translational position of the optical elements.

Suitable mechanical and/or electrical components can be provided forcontrolling and driving the relative rotation of the cylindricalelements. The spot size depends nonlinearly of the rotation angle α, asdiscussed in the detailed analysis of the configurations presented.Therefore, a fine rotation mechanism with good reproducibility andideally no backlash is desirable. A rotating ring rigidly attached tothe rotating lens/mirror holder can suffice. The holder should be wellcentered to the mechanical axis of the whole device, which should becoincident with the optical axis. An elastic washer can be provided toallow the necessary friction such as to avoid losing the desiredposition of the ring during operation, and therefore to keep the spotsize at the desired value. If desired, several discrete angularpositions corresponding to several discrete magnifications can beselected by appropriate ring indentations. A mechanical stop to limitthe whole rotation range to 90 degree can be used. Thin line graduationson the outer tube and a mark on the rotating ring can also be used tocontrol the angular position of the lens(es). More precise solutions mayuse gears to control the rotation of the ring, and stepper motors drivenby appropriate electronics and computer software. Special software canbe implemented to allow the precise control of the spot size by using aspot size optical detector (based, for example on pinhole/quadrantdetectors) which will give a feedback signal to the computer to controland maintain the spot size at the desired value. Fine rotation control,with approximately 1 minute of angle resolution and reproducibility(especially when small spots are desired to be controlled in focus-modeconfigurations) can be obtained by using levers attached to the rotationring and activated by fine translation mechanisms, as micrometric ordifferential screws. A focus-mode configuration coupled with a smallspot size measuring system can be used as a sensor for very smallangular movements, with a resolution in the range of 1 second of arc,for a limited range (several degrees) of angular movements.

FIG. 11 shows a side sectional view of a mounting/control arrangementsuitable for controlling the operation of a three-lens system asdescribed above. An optical system 920 includes a first cylindrical lens924 rigidly mounted within a protective housing 980. Lens 924 is mountedwithin a fixed cylindrical lens holder 982, which is in turn fixedlyattached to housing 980. A block holder 984 is fixedly mounted withinlens holder 982. The translational position of block holder 984 isadjusted during the assembly of system 920 to ensure that a secondcylindrical lens 926 is positioned at a desired distance (2 f) away fromfirst lens 924. Once block holder 984 is positioned at the desiredlocation, it is affixed to lens holder 982 by inserting screws through aset of holes 994 defined in the wall of lens holder 982, and pressingthe screws to the surface of block holder 984.

Second cylindrical lens 926 and an attached spherical lens 960 arerigidly mounted on a rotatable lens holder 986. The translationalposition of lens holder 986 is held fixed by a locking flange 988 whichholds lens holder 986 against block holder 984. A rotatable handlingring 990 is rigidly coupled to lens holder 986, for coupling an externalrotational driving force to lens holder 986 so as to rotate second lens926.

Cylindrical lenses 924, 926 are identical, and have a focal length f=50mm. The spherical lens 960 has a focal length f₀=140 mm. The device actson a 3–4 mm diameter collimated, low power laser beam (red laser diode),and provides at the working distance a round spot continuouslyadjustable from a minimum of approximately 50 μm to a maximum ofapproximately 12 mm.

Generally, the first cylindrical lens or other cylindrical element asdescribed above need not be provided within the same housing as theother elements/lenses. The function performed by the first cylindricalelement may be performed by outside components, for example by aseparately-mounted lens, or by components present within the laser usedto generate the beam of interest. Such components need not necessarilyinclude a cylindrical lens.

Qualitatively, the first cylindrical element/lens in the systemsdescribed above induces a desired astigmatism in an incoming stigmaticbeam. The second cylindrical element is placed in the round spot of thebeam with astigmatism emerging after the first cylindrical element. Alaser may be used to generate a beam with some astigmatism. Such anastigmatism may arise from the optics inside a laser head. A laser headis a box containing the active medium emitting the laser radiation,associated optics, a system delivering energy to the laser medium, athermal stabilization system, and other components. Such internal opticsmay, but need not, include a cylindrical lens. The overall effect of alloptical components, including the laser cavity mirrors, may be replacedby an equivalent cylindrical lens, even though a physical cylindricallens is not present. Such a beam with classical astigmatism always hasat least one round spot, real or virtual, i.e. inside the laser head,and not accessible to be detected on a screen (virtual), or outside thelaser system, accessible to be detected on a screen (real).

A virtual round spot can be transformed into a real round spot situatedoutside the laser head and accessible on a screen by using a relayspherical lens acting as an imaging system. The virtual round spot is avirtual object for the spherical lens. The virtual object can betransformed into a real image after the lens, provided an appropriateoptical power and lens position are selected.

A real round spot, situated outside the laser head, of a beam withastigmatism is accessible to be processed with an appropriatecylindrical lens. The real round spot can be transformed at somedistance after the cylindrical lens into an adjustable round spot, byrotating the cylindrical lens. Such a single cylindrical lenscorresponds to the second cylindrical lens of the two-lens systemsdescribed above. Different single-lens configurations corresponding tothe two-lens configurations described above can be generated:focus-mode, image-mode, and projection-mode. To generate suchconfigurations, a single rotating cylindrical lens is placed at theplane where a real round spot of an astigmatic beam is located.

It will be clear to one skilled in the art that the above embodimentsmay be altered in many ways without departing from the scope of theinvention. For example, cylindrical lenses include plano-convex andplano-concave cylindrical lenses, as well as lenses having twocylindrical surfaces, such as compound-cylindrical (or bi-cylindrical)lenses acting along two orthogonal axes. Accordingly, the scope of theinvention should be determined by the following claims and their legalequivalents.

1. An optical method comprising: generating a symmetrically-scalable spot on a target positioned at a working location by passing a collimated, stigmatic light beam sequentially through a first cylindrical lens and a second cylindrical lens of the same sign as the first cylindrical lens, the second cylindrical lens being positioned adjacent to the first cylindrical lens such that the light beam incident on the first cylindrical lens maintains a substantially round spot across the first cylindrical lens and the second cylindrical lens; and symmetrically scaling the spot at the working location by adjusting an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens by rotating at least one of the first cylindrical lens and the second cylindrical lens.
 2. The method of claim 1, wherein the first cylindrical lens and the second cylindrical lens are positive cylindrical lenses.
 3. The method of claim 2, wherein the first cylindrical lens and the second cylindrical lens have equal focal lengths.
 4. The method of claim 1, wherein the first cylindrical lens and the second cylindrical lens are negative cylindrical lenses.
 5. The method of claim 4, wherein the first cylindrical lens and the second cylindrical lens have equal focal lengths.
 6. The method of claim 1, comprising generating the symmetrically scalable spot by illuminating an imaged object, and further comprising rotating the imaged object in tandem with the at least one of the first cylindrical lens and the second cylindrical lens, to symmetrically scale the spot at the working location without rotating the spot at the working location.
 7. The method of claim 1, wherein a distance between the first cylindrical lens and the second cylindrical lens is less than or equal to about 2 mm.
 8. An optical system comprising: a first cylindrical lens; and a second cylindrical lens of the same sign as the first cylindrical lens, the second cylindrical lens being positioned adjacent to the first cylindrical lens such that a collimated and stigmatic light beam incident on the first cylindrical lens maintains a substantially round spot across the first cylindrical lens and the second cylindrical lens; wherein at least one of the first cylindrical lens and the second cylindrical lens is rotatable so as to adjust an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens to symmetrically scale a spot generated by the light beam at a target location.
 9. The optical system of claim 8, wherein the first cylindrical lens and the second cylindrical lens are positive cylindrical lenses.
 10. The optical system of claim 9, wherein the first cylindrical lens and the second cylindrical lens have equal focal lengths.
 11. The optical system of claim 8, wherein the first cylindrical lens and the second cylindrical lens are negative cylindrical lenses.
 12. The optical system of claim 11, wherein the first cylindrical lens and the second cylindrical lens have equal focal lengths.
 13. The optical system of claim 8, further comprising: a projection screen positioned at the target location; and an imaged object rotatable in tandem with the at least one of the first cylindrical lens and the second cylindrical lens, the symmetrically scalable spot being generated by illuminating the imaged object.
 14. The optical system of claim 8, wherein a distance between the first cylindrical lens and the second cylindrical lens is less than or equal to about 2 mm.
 15. An optical system comprising: a first cylindrical lens; a second cylindrical lens of the same sign as the first cylindrical lens, the second cylindrical lens being positioned adjacent to the first cylindrical lens; and a rotation device coupled to at least one of the first cylindrical lens and the second cylindrical lens, for rotating the at least one of the first lens and the second lens about an optical axis to adjust an angle between a principal axis of the first lens and a principal axis of the second lens, thereby controlling a focusing of a light beam passing through the first cylindrical lens and the second cylindrical lens.
 16. An optical method comprising: generating a symmetrically-scalable spot on a target positioned at a working location by passing a collimated and stigmatic light beam sequentially through a first cylindrical lens and a second cylindrical lens, the first cylindrical lens and the second cylindrical lens having focal lengths of opposite sign and unequal magnitude, the second cylindrical lens being positioned next to the first cylindrical lens such that the light beam incident on the first cylindrical lens maintains a substantially round spot across the first cylindrical lens and the second cylindrical lens; and symmetrically scaling the spot at the working location by adjusting an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens by rotating at least one of the first cylindrical lens and the second cylindrical lens.
 17. The method of claim 16, further comprising passing the light beam through a spherical lens after passing the light beam through the first cylindrical lens and the second cylindrical lens.
 18. The method of claim 16, comprising generating the symmetrically scalable spot by illuminating an imaged object, and further comprising rotating the imaged object in tandem with the at least one of the first cylindrical lens and the second cylindrical lens, to symmetrically scale the spot at the working location without rotating the spot at the working location.
 19. The method of claim 16, wherein a distance between the first cylindrical lens and the second cylindrical lens is less than or equal to about 2 mm.
 20. An optical system comprising: a first cylindrical lens; and a second cylindrical lens positioned next to the first cylindrical lens such that a collimated, stigmatic light beam incident on the first cylindrical lens maintains a substantially round spot across the first cylindrical lens and the second cylindrical lens, the first cylindrical lens and the second cylindrical lens having focal lengths of opposite sign and unequal magnitude; wherein at least one of the first cylindrical lens and the second cylindrical lens is rotatable so as to adjust an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens to symmetrically scale a spot generated by the light beam at a target location.
 21. The optical system of claim 20, further comprising a spherical lens positioned to receive the light beam after passage through the first cylindrical lens and the second cylindrical lens.
 22. The optical system of claim 20, further comprising: a projection screen positioned at the target location; and an imaged object rotatable in tandem with the at least one of the first cylindrical lens and the second cylindrical lens, the symmetrically scalable spot being generated by illuminating the imaged object.
 23. The optical system of claim 20, wherein a distance between the first cylindrical lens and the second cylindrical lens is less than or equal to about 2 mm.
 24. An optical system comprising: a triplet of physically-adjacent lenses, comprising a first cylindrical lens having a first focal length, for receiving a collimated light beam, a second cylindrical lens having a second focal length equal and opposite in sign to the first focal length, for receiving the beam after passage through the first cylindrical lens; and a convergent spherical lens having a third focal length; wherein the second cylindrical lens is rotatable with respect to the first cylindrical lens so as to adjust an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens, to generate a symmetrically-scalable spot on a target situated away from the spherical lens.
 25. The optical system of claim 24, wherein the symmetrically-scalable spot is generated at a distance substantially equal to the third focal length away from the spherical lens.
 26. The optical system of claim 24, further comprising the target, wherein the target comprises a projection screen.
 27. The optical system of claim 24, further comprising a light source for generating the light beam.
 28. The optical system of claim 27, wherein the light beam is stigmatic.
 29. The optical system of claim 24, wherein the lenses of the triplet of physically-adjacent lenses are separated by distances less than or equal to about 2 mm.
 30. The optical system of claim 24, wherein the third focal length is equal in magnitude to the first focal length.
 31. An optical method comprising: generating a symmetrically-scalable spot on a target positioned at a working location by passing a collimated, intrinsic stigmatic light beam through a triplet of physically adjacent lenses comprising a first cylindrical lens having a first focal length, a second cylindrical lens having a second focal length, and a spherical lens having a third focal length, wherein the second focal length is equal and opposite in sign to the first focal length; and symmetrically-scaling the spot at the working location by adjusting an angle between a principal axis of the first cylindrical lens and a principal axis of the second cylindrical lens by rotating at least one of the first cylindrical lens and the second cylindrical lens.
 32. The method of claim 31, wherein the symmetrically-scalable spot is generated at a distance substantially equal to the third focal length away from the spherical lens.
 33. The method of claim 31, comprising generating the symmetrically scalable spot by illuminating an imaged object, and further comprising rotating the imaged object in tandem with the at least one of the first cylindrical lens and the second cylindrical lens, to symmetrically scale the spot at the working location without rotating the spot at the working location.
 34. The method of claim 31, wherein the light beam is stigmatic.
 35. The method of claim 31, wherein the lenses of the triplet of physically-adjacent lenses are separated by distances less than or equal to about 2 mm.
 36. An optical method comprising: passing a light beam with astigmatism through a rotatable cylindrical lens positioned at a round spot location along the light beam; and rotating the cylindrical lens to symmetrically-scale a spot at a working distance away from the cylindrical lens.
 37. The method of claim 36, further comprising passing the light beam through a spherical lens before passing the light beam through the cylindrical lens.
 38. An optical system comprising: a light source for generating a light beam with astigmatism; a rotatable cylindrical lens optically coupled to the light source, for receiving the light beam, the cylindrical lens being positioned at a round spot location along the light beam; and a rotation device coupled to the cylindrical lens, for rotating the cylindrical lens to symmetrically-scale a spot at a working distance away from the cylindrical lens.
 39. The optical system of claim 38, further comprising a spherical lens positioned between the light source and the cylindrical lens. 